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Full text of "Scientific Papers - Vi"

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150 THE COKRECTION TO THE LENGTH OF the simplification arising from the fact that (1) is practically a member c series qf>. The calculated capacity, an overestimate unless all the coefficients 1 correctly assigned, is given by addition of (16) and (24). The first apj mation is obtained by omitting all the quantities H, so that the B's vanisl The additional capacity, derived entirely from (16), is then ffiSkAW^k on introduction of the value of A, log2 b/a ^ ttbTJ-T(kb)' .........'•..............(' the summation extending to all the roots of /„ (kb) = 0. Or if we ex the result in terms of the correction 81 to the length (for one end), we hi .. 26 ^ J,?(ka) 81 — asjjhe first approximation to 81 and an overestimate. The series in (26) converges sufficiently. Jo2 (ka) is less than unity. mth root of </„ (so) — 0 is as — (m —\}TT approximately, and J^2 (as) = 2/7 that when m is great The values' of the reciprocals of (x?J-? (so) for the earlier roots can be calcr from the tables* and for the higher roots from (27). I find • • • m X ± Ji (*) ar3 -f- JjS (a;) 1 ....... .. S'4048 •51915 •2668 2 ......... 5-5201 •34027 •0513 3 ......... 8-6537 •27145 •0209 4 ......... 11-7915 •23245 •0113 5 ......... 14-9309 •20655 •0070 The next five values are '0048., '0035; "0026, "0021, '0017. Thus fo: value of a the series in (26) is •2668 Jo"(2-405 a/6) + -0513 J02 (5'520 a/6) + ...; ......( it can be calculated without difficulty when a/b is given. When a/b is small, the J's in (28) may be omitted, and we have simply to sum the nu in the-fourth column of'the table and its continuation. The first ten give '3720. The remainder I estimate at '015, making in all '387. T] this case gz= -7746 ~ Iog6/a............................... .* Gray and Mathews, Vessel's Functions, pp. 244, 247.nts, even though only a limited number included. Every fresh coefficient that is included renders the approximate closer, and as near an approach as we please to the truth may be arrived by continuing the process. The true value of (14) is equal by Gree theorem to